Summary: We survey the set of computational methods devised for implementing the MiSaTaQuWa formulation in practice, for orbits around Kerr black holes. We focus on the gravitational self-force (SF) and review in detail two of these methods: (i) the standard mode-sum method, in which the perturbation field is decomposed into multipole harmonics and the MiSaTaQuWa regularization is performed, effectively, mode by mode; and (ii) m-mode regularization, whereby one regularizes individual azimuthal modes of the full perturbation. The implementation of these strategies involves the numerical integration of the relevant perturbation equations, and we discuss several practical issues that arise and ways to deal with them. These issues include the choice of gauge, the numerical representation of the particle singularity, and the handling of high-frequency contributions near the particle in frequency-domain calculations. As an example, we show results from an actual computation of the gravitational SF for an eccentric geodesic orbit around a Schwarzschild black hole, using direct numerical integration of the Lorenz-gauge perturbation equations in the time domain.
For the entire collection see [Zbl 1209.83003].